Theorem frege56b

Description: Lemma for frege57b . Proposition 56 of Frege1879 p. 50. (Contributed by RP, 24-Dec-2019) (Proof modification is discouraged.)

	Ref	Expression
Assertion	frege56b	⊢ ( ( 𝑥 = 𝑦 → ( [ 𝑥 / 𝑧 ] 𝜑 → [ 𝑦 / 𝑧 ] 𝜑 ) ) → ( 𝑦 = 𝑥 → ( [ 𝑥 / 𝑧 ] 𝜑 → [ 𝑦 / 𝑧 ] 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege55b	⊢ ( 𝑦 = 𝑥 → 𝑥 = 𝑦 )
2		frege9	⊢ ( ( 𝑦 = 𝑥 → 𝑥 = 𝑦 ) → ( ( 𝑥 = 𝑦 → ( [ 𝑥 / 𝑧 ] 𝜑 → [ 𝑦 / 𝑧 ] 𝜑 ) ) → ( 𝑦 = 𝑥 → ( [ 𝑥 / 𝑧 ] 𝜑 → [ 𝑦 / 𝑧 ] 𝜑 ) ) ) )
3	1 2	ax-mp	⊢ ( ( 𝑥 = 𝑦 → ( [ 𝑥 / 𝑧 ] 𝜑 → [ 𝑦 / 𝑧 ] 𝜑 ) ) → ( 𝑦 = 𝑥 → ( [ 𝑥 / 𝑧 ] 𝜑 → [ 𝑦 / 𝑧 ] 𝜑 ) ) )